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Question

Let A,B,C be points with position vectors 2^i^j+^k,^i+2^j+^k and 3^i+^j+2^k respectively. Find the shortest distance between point B and plane OAC.

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Solution

Given points are,

A(2,1,1),B(1,2,1),C(3,1,2)

Let the given points be

x1=0,y1=0,z1=0x2=2,y2=1,z2=1x3=3,y3=1,z3=2

Then ,

equation of plane passing through O,A,C is

∣ ∣xx1yy1zz1x2x1y2y1z2z1x3x1y3y1z3z1∣ ∣=0∣ ∣xyz211312∣ ∣=0x(21)y(43)+z(2+3)=03x+y5z=0

Now,
Shortest distance between point B(1,2,1) and plane OAC is

d=∣ ∣ ∣3×1+2×15×132+12+(5)2∣ ∣ ∣=0

So, the point B is at no distance from the plane and hence it lies on the plane.

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